Part (1) is [1]}:
it is clear that any subset \(U \subset M\)  which is open in the manifold topology is also open in the diffeological topology.
Conversely, if \(U\)  is open in \(\mathrm {D}(\underline{M})\) , then its intersection with all images of charts of \(M\)  must be open.
As these images form a basis for the manifold topology, \(U\)  is open in the manifold topology.
